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To cite this version:
Denis Roegel. A reconstruction of Peters’s table of 7-place logarithms (volume 1, 1940). [Research Re- port] LORIA, UMR 7503, Université de Lorraine, CNRS, Vandoeuvre-lès-Nancy. 2016. �hal-01357840�
Peters’s table
of 7-place logarithms (volume 1, 1940)
Denis Roegel
29 August 2016
This document is part of the LOCOMAT project:
http://locomat.loria.fr
basis of many later tables, most of which have been reconstructed by us.1
The 8-place table was itself based on a 12-place manuscript table, which was again used by Peters in the preparation of a new 7-place table of logarithms of trigonometrical functions for every second of the quadrant published in 1911 [50].
In 1919 and 1922, Peters published a 10-place table of logarithms [56, 55, 59], and 6 and 7-place tables of logarithms of trigonometrical functions at 0◦.001 intervals in 1921 [57, 58].
The present table comprises two volumes and was published in 1940 [70]. Here we describe only the first volume, together with its reconstruction. The second volume, covering the logarithms of trigonometrical functions, is described separately [99]. These two volumes were derived from the earlier 10-place tables [10].
2 Description of the tables
The first volume published in 1940 contains the logarithms of the integers from 1 to 1000 and from 10000 to 100000, the antilogarithms from 00000 to 100000, and finally addition (A) and subtraction (S) logarithms, all to seven places.
These two volumes of Peters’s tables were reviewed by Archibald [10].
2.1 Logarithms
There are many tables of logarithms, and we direct the reader to the introduction of Bauschinger and Peters’s 8-place table [17] for some references, or to other reconstructions found on LOCOMAT (locomat.loria.fr) for more comprehensive lists.
The main 7-place table of logarithms is that published by Edward Sang in 1871 [109].
It gave the logarithms of all numbers from 20000 to 200000. Its range is therefore more extensive than that of Peters’s table.
Peters’s table was derived from the earlier 10-place tables published in 1922 [59].
Only 99 numbers could not readily be abridged. For 96 of these numbers, the correct abridgement was obtained using the 12-place manuscript table. That table gave the logarithms of numbers with at most one unit of error on the 12th place. The three remaining numbers were recomputed using well known methods. Peters mentions the following series
logx= 1
2log(x+ 1) + 1
2log(x−1) +M
1
2x2 −1 + 1
3(2x2−1)+· · ·
(1) but his words do not seem to mean that it was used, or that only this formula was used for the three remaining numbers.
1For more information on Peters’s tables, we refer the reader to our summary [107].
3
2.2 Antilogarithms
Extensive tables of antilogarithms have also been published before. We mention here only the most important such tables. In 1742, Dodson [29] published 300 pages of tables giving the antilogarithms from 0.00000 to 1.00000, to 11 places. In 1844 and 1849, Shortrede published a 7-place table of antilogarithms [113] (with interval 0.00001), as did Filipowski [31] in 1849 (also with an interval of 0.00001). Peters’s table of antilogarithms is therefore identical in scope and accuracy to those of Shortrede and Filipowski.
Peters, however, did not borrow the values from Shortrede or Filipowski. Instead, as the basis of his calculations, Peters used Prytz’s table of antilogarithms published in 1886 [77]. This table contains 15-place antilogarithms for mantissas from 000 to 999 at unit intervals. Peters took a 12-place excerpt and using fourth differences, he interpolated intermediate values, so that he had antilogarithms from 000 to 999 at intervals of 0.5 units of the argument. This led to a 12-place table of antilogarithms from 00000 to 100000, at intervals of 50. Then, 49 values were interpolated between two such antilogarithms, so that a 12-place table of antilogarithms from 00000 to 100000, at intervals of 1, was produced. The errors in this table are always less than 0.502 units of the seventh place.
By comparison with Shortrede’s table (probably the one published in 1844 [113], but Peters mentions a non-existing edition from 1854), the error on the seventh place could be made no larger than 0.5 units of the seventh place.
In the preface of his work, Peters gives a number of reasons why tables of antiloga- rithms are more useful than tables of logarithms.
2.3 Addition and subtraction logarithms
Addition and subtraction logarithms were first introduced by Leonelli in 1803 [46], pop- ularized by Gauss [34], and there have been a number of large tables of such loga- rithms [47, 121, 18, 128, 114, 127]. A table which appeared just before Peters’s was Cohn’s 6-place table [24, 30].
These logarithms are sometimes called “Gaussian logarithms.” Summaries of these logarithms are given by Glaisher [36], Archibald [8], Kühn [44] and others.
The idea of these logarithms is merely to introduce two functions A and S such that log(a+b) = loga+A(loga,logb) (2) log(a−b) = loga−S(loga,logb) (3) It is easy to derive expressions of these two functions, setting D = loga−logb. The functions A and S actually depend only onD, and we have
A(D) = log(1 + 10−D) (4)
S(D) = −log(1−10−D) (5)
4
Compared with the above formulæ, Peters’s tables for subtraction logarithms exhibit a further refinement, in that S(S(x)) = x, and therefore S(x) = S−1(x), which means that Peters could only give part of the table of S. This is why the argument of S goes from D = 0.3 to D = 6.99, with values of S ranging from 0.3020624 to 0, but that backwards, the same table will give the values of S for D ranging from 0.3020624 to 0.
Therefore, this table gives the values of S forD ranging from0 to6.99, exactly the same range as the table for addition logarithms. In using subtraction logarithms, one has to find out if one is using the table from the argument to the value, or conversely, depending if D is greater or smaller than 0.3.
As a simple example demonstrating the use of these tables, consider a = 3.01, b = 2.38, then loga = 0.4785665, logb = 0.3765770, D = 0.1019895, and the tables give A(D) = 0.2530618 and S(D) = 0.6792 (obtained by looking up the value of D in the values of S as described above, here without interpolation), from which we obtain:
log(a+b) = loga+ 0.2530618 = 0.4785665 + 0.2530618 = 0.7316283 (6)
log(a−b) = loga−0.6792 = 0.7994−1 (7)
We can check that these logarithms lead to values of a+b and a−b which are 5.390 and 6.301/10 = 0.6301, nearly the values we started with. This example incidentally highlights that the purpose of the addition and subtraction logarithms is not to compute a+b or a−b, which does not benefit from logarithms, but only to compute log(a+b) and log(a−b) which may be useful in other computations.
One should be aware that there have been different definitions of these logarithms.
For instance Wittstein [127] defines functionsA0 and S0 such thatlog(a+b) = logb+A0 and log(a−b) = logb+S0, and A0 and S0 are not the same functions as Peters’sA and S:
A0(D) = log(1 + 10D) (8)
S0(D) = log(10D −1) (9)
But the purpose of these functions remains the same.
Peters’s tables of addition and subtraction logarithms are based on the fundamental tables published in 1922 by Andoyer [4]. These tables gave 16-place values of A and S, and Peters used them to develop 10-place tables, whose error on the seventh place was less than 0.502. This 10-place table was computed by interpolation, the details being given in Peters’s introduction, and being similar to other interpolations performed by Peters. Eventually, a number of values had to be recomputed, in order to ensure that the error on the final table was no larger than 0.5 units of the seventh place.
Archibald mentions that Peters had completed an 8-place table of addition and sub- traction logarithms [8, p. 114], but we do not know if that table is the result of a parallel project, also based on Andoyer’s 16-place tables.
5
of a line and the first value of the next one. The range of these differences may differ from those given in the lower part of the pages, in particular because there may be larger or smaller differences between the first and second values of a page, or between the penultimate and last values of a page. There are also a number of cases where there was not sufficient room to give all possible differences, and the ranges are then not continuous.
The first volume is supplemented by a collection of formulæ and constants, series, plane and spherical trigonometry, etc.
6
[1] ???? On the eight-figure table of Peters and Comrie. Mathematical Tables and other Aids to Computation, 1(2):64–65, 1943. [The title is ours, and there are actually two notices, on the accuracy of the table published in 1939 [69], and its comparison with other tables.]
[2] Marie Henri Andoyer. Nouvelles tables trigonométriques fondamentales contenant les logarithmes des lignes trigonométriques. . .. Paris: Librairie A. Hermann et fils, 1911. [Reconstruction by D. Roegel in 2010 [81].]
[3] Marie Henri Andoyer. Nouvelles tables trigonométriques fondamentales contenant les valeurs naturelles des lignes trigonométriques. . .. Paris: Librairie A. Hermann et fils, 1915–1918. [3 volumes, reconstruction by D. Roegel in 2010 [82].]
[4] Marie Henri Andoyer. Tables fondamentales pour les logarithmes d’addition et de soustraction. Bulletin astronomique, 2:5–32, 1922.
[5] Raymond Clare Archibald. J. T. Peters, Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten. Mathematical Tables and other Aids to Computation, 1(1):11–12, 1943. [review of the edition published in 1939 [69]]
[6] Raymond Clare Archibald. J. T. Peters, Seven-place values of trigonometric functions for every thousandth of a degree. Mathematical Tables and other Aids to Computation, 1(1):12–13, 1943. [review of the edition published in 1942 [54]]
[7] Raymond Clare Archibald. Tables of trigonometric functions in non-sexagesimal arguments. Mathematical Tables and other Aids to Computation, 1(2):33–44, 1943.
[8] Raymond Clare Archibald. Theodor Wittstein, Addition and Subtraction Logarithms to Seven Decimal Places. Mathematical Tables and other Aids to Computation, 1(4):112–114, 1943. [review of the 1943 reprint of Wittstein’s table [127]]
[9] Raymond Clare Archibald. J. T. Peters, Eight-place table of trigonometric functions for every sexagesimal second of the quadrant. Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten.
2Note on the titles of the works: Original titles come with many idiosyncrasies and features (line splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore seemed pointless to capitalize works according to conventions which not only have no relation with the original work, but also do not restore the title entirely. In the following list of references, most title words (except in German) will therefore be left uncapitalized. The names of the authors have also been homogenized and initials expanded, as much as possible.
The reader should keep in mind that this list is not meant as a facsimile of the original works. The original style information could no doubt have been added as a note, but we have not done it here.
12
the edition published in 1940 [70]]
[11] Raymond Clare Archibald. Johann Theodor Peters. Mathematical Tables and other Aids to Computation, 1(5):168–169, 1944. [obituary notice]
[12] Raymond Clare Archibald. J. T. Peters, Sechsstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades. Mathematical Tables and other Aids to Computation, 2(19):298–299, 1947. [review of 9th edition of [68]
published in 1944]
[13] Raymond Clare Archibald. J. T. Peters, Siebenstellige Werte der
trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades.
Mathematical Tables and other Aids to Computation, 2(19):299, 1947. [review of the 1941 edition [71]]
[14] Raymond Clare Archibald. Mathematical table makers. Portraits, paintings, busts, monument. Bio-bibliographical notes. New York: Scripta Mathematica, 1948.
[contains a photograph of Peters]
[15] Julius Bauschinger. Interpolation. In Wilhelm Franz Meyer, editor,Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, volume 1(2), pages 799–820. Leipzig: B. G. Teubner, 1904. [a French translation appeared in [111]]
[16] Julius Bauschinger and Johann Theodor Peters. Logarithmic-trigonometrical tables with eight decimal places etc. Leipzig: Wilhelm Engelmann, 1910–1911. [2 volumes, English introduction. See [17] for the German edition.]
[17] Julius Bauschinger and Johann Theodor Peters. Logarithmisch-trigonometrische Tafeln mit acht Dezimalstellen etc. Leipzig: Wilhelm Engelmann, 1910–1911. [2 volumes, German introduction. See [16] for the English edition; these volumes have been reprinted in 1936, 1958 and 1970, but the introductions vary. In particular, details of the construction of Hamann’s machine were dropped in the last editions. Reconstructions are given in [87] and [88].]
[18] J. E. Boner. Die Logarithmen und die Gränzen ihrer Zuverlässlichkeit, die Gaussischen Logarithmen für Summen und Differenzen und zur logarithmischen Auflösung der quadratischen Gleichungen. Münster: Friedr. Regensberg, 1842.
[19] Henry Briggs. Arithmetica logarithmica. London: William Jones, 1624. [The tables were reconstructed by D. Roegel in 2010. [84]]
[20] Henry Briggs and Henry Gellibrand. Trigonometria Britannica. Gouda: Pieter Rammazeyn, 1633. [The tables were reconstructed by D. Roegel in 2010. [83]]
13
Astronomischen Gesellschaft, 39:158, 232–240, 1904.
[23] Berthold Cohn. Tafeln der Additions- und Subtraktions-Logarithmen auf sechs Dezimalen. Leipzig: Wilhelm Engelmann, 1909. [not seen, second edition in 1939 [24]]
[24] Berthold Cohn. Tables of addition and subtraction logarithms with six decimals.
London: Scientific computing services, 1939. [not seen, first edition in 1909 [23]]
[25] Leslie John Comrie. Logarithmic and trigonometrical tables. Monthly Notices of the Royal Astronomical Society, 85(4):386–388, 1925. [mentions several of Peters’s tables]
[26] Leslie John Comrie. J. T. Peters, Sechsstellige Tafel der trigonometrischen Funktionen,. . . , Berlin, 1929. Mathematical Tables and other Aids to
Computation, 1(5):162, 1944. [Attributes errors in the first edition of [60] to one of the proofreaders of the table.]
[27] Charles H. Cotter. Gaussian logarithms and navigation. Journal of Navigation, 24(4):569–572, 1971.
[28] Harold Thayer Davis, editor. Tables of the higher mathematical functions.
Bloomington, In.: The principia press, Inc., 1933, 1935. [2 volumes]
[29] James Dodson. The anti-logarithmic canon. Being a table of numbers consisting of eleven places of figures, corresponding to all logarithms under 100000. London, 1742.
[30] W. A. F. B. Cohn, Addition and Subtraction Logarithms with Six Decimals.
Journal of the Institute of Actuaries, 70(2):260–261, July 1939. [review of Cohn’s table [24]]]
[31] Herschell E. (Z.evi Hirsch) Filipowski. A table of anti-logarithms; containing to seven places of decimals, natural numbers, answering to all logarithms from
·00001 to ·99999, and an improved table of Gauss’s logarithms, by which may be found the logarithm to the sum or difference of two quantities whose logarithms are given. Etc. London: M. & W. Collis, 1849.
[32] Alan Fletcher, Jeffery Charles Percy Miller, and Louis Rosenhead. An index of mathematical tables. London: Scientific computing service limited, 1946.
[33] Alan Fletcher, Jeffery Charles Percy Miller, Louis Rosenhead, and Leslie John Comrie. An index of mathematical tables (second edition). Reading, Ma.:
Addison-Wesley publishing company, 1962. [2 volumes]
14
[35] Erwin Gigas. Professor Dr. Peters und sein Werk. Nachrichten aus dem Reichsvermessungsdienst. Mitteilungen des Reichsamts für Landesaufnahme, 17:346–350, 1941.
[36] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.
London: Taylor and Francis, 1873. [Also published as part of the “Report of the forty-third meeting of the British Association for the advancement of science,” London: John Murray, 1874. A review by R. Radau was published in theBulletin des sciences mathématiques et astronomiques, volume 11, 1876, pp. 7–27]
[37] James Henderson. Bibliotheca tabularum mathematicarum, being a descriptive catalogue of mathematical tables. Part I: Logarithmic tables (A. Logarithms of numbers), volume XIII of Tracts for computers. London: Cambridge University Press, 1926.
[38] Samuel Herrick, Jr. Natural-value trigonometric tables. Publications of the Astronomical Society of the Pacific, 50(296):234–237, 1938.
[39] Peter Holland. Biographical notes on Johann Theodor Peters, 2011.
www.rechnerlexikon.de/en/artikel/Johann_Theodor_Peters
[40] Wilhelm Rudolf Alfred Klose. Prof. Dr. Jean Peters gestorben. Zeitschrift für Angewandte Mathematik und Mechanik, 22(2):120, 1942. [obituary notice]
[41] Otto Kohl. Jean Peters. Vierteljahresschrift der Astronomischen Gesellschaft, 77:16–20, 1942. [includes one photograph]
[42] August Kopff. Jean Peters†. Astronomische Nachrichten, 272(1):47–48, 1941.
[43] Christine Krause. Das Positive von Differenzen : Die Rechenmaschinen von Müller, Babbage, Scheutz, Wiberg, . . . , 2007.
[44] Klaus Kühn. C. F. Gauss und die Logarithmen. Mitteilungen der Gauss-Gesellschaft, 40:75–83, 2003.
[45] A. V. Lebedev and R. M. Fedorova. A guide to mathematical tables. Oxford:
Pergamon Press, 1960.
[46] Zecchini Leonelli. Supplément logarithmique, contenant la décomposition des grandeurs numériques quelconques en facteurs finis ; reconnue très-propre, et incomparablement plus courte que toute autre méthode, pour calculer directement les logarithmes et leurs valeurs naturelles, à l’aide des logarithmes de ces facteurs, et munie de trois tables de logarithmes facteurs : les deux premières pour les
15
A. Brossier, 1803.
[47] Erhard Adolph Matthiessen. Tafel zur bequemern Berechnung des Logarithmen der Summe oder Differenz zweyer Größen, welche selbst nur durch ihre
Logarithmen gegeben sind. Altona: Johann Friederich Hammerich, 1817.
[48] Johann Theodor Peters. Neue Rechentafeln für Multiplikation und Division mit allen ein- bis vierstelligen Zahlen. Berlin: G. Reimer, 1909. [also published in 1919 and 1924 by Walter de Gruyter & Co.; the library of the Paris observatory also has a variant of the 1909 edition with the French title “Nouvelles tables de calcul pour la multiplication et la division de tous les nombres de un à quatre chiffres” (as well as a French introduction), which the library kindly checked for us; and the 1924 edition seems to be an English one with the title
“New calculating tables for multiplication and division by all numbers of from one to four places.” We have only had the 1919 edition in hands, and we reconstructed it in [104].]
[49] Johann Theodor Peters. Einundzwanzigstellige Werte der Funktionen Sinus und Cosinus : zur genauen Berechnung von zwanzigstelligen Werten sämtlicher trigonometrischen Funktionen eines beliebigen Arguments sowie ihrer
Logarithmen. Berlin: Verlag der Königl. Akademie der Wissenschaften, 1911. [54 pages, Appendix 1 to the “Abhandlungen der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse.”, not seen, but reprinted at the end of the English edition of [69]]
[50] Johann Theodor Peters. Siebenstellige Logarithmentafel der trigonometrischen Funktionen für jede Bogensekunde des Quadranten. Leipzig: Wilhelm Engelmann, 1911. [reconstructed in [94]]
[51] Johann Theodor Peters. Fünfstellige Logarithmentafel der trigonometrischen Funktionen für jede Zeitsekunde des Quadranten. Berlin: Reimer, 1912.
[reconstructed in [92]]
[52] Johann Theodor Peters. Tafeln zur Berechnung der Mittelpunktsgleichung und des Radiusvektors in elliptischen Bahnen für Excentrizitätswinkel von 0◦ bis 24◦. Berlin: Ferd. Dümmler, 1912. [second edition in 1933]
[53] Johann Theodor Peters. Dreistellige Tafeln für logarithmisches und numerisches Rechnen. Berlin: P. Stankiewicz, 1913. [not seen, second edition in 1948 (seen),
reconstructed in [89]]
[54] Johann Theodor Peters. Siebenstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Grades. Berlin-Friedenau: Verlag der Optischen Anst. Goerz, 1918. [Reprinted in 1938 and 1941, as well as in 1942 in English with the title “Seven-place Values of trigonometric functions for every thousandth of a degree.”, all four editions seen. Reconstructed in [95].]
16
Logarithmen der trigonometrischen Funktionen von 0 bis 90 für jedes Tausendstel des Grades. Berlin: Reichsamt f. Landesaufnahme, 1919. [not seen, second edition in 1957 (seen); also Russian editions in 1964 and 1975; reconstructed in [106]]
[57] Johann Theodor Peters. Sechsstellige Logarithmen der trigonometrischen Funktionen von 0◦ bis 90◦ für jedes Tausendstel des Grades. Berlin: Verlag der preussischen Landesaufnahme, 1921. [reconstructed in [102]]
[58] Johann Theodor Peters. Siebenstellige Logarithmen der trigonometrischen Funktionen von 0◦ bis 90◦ für jedes Tausendstel des Grades. Berlin: Verlag der preussischen Landesaufnahme, 1921. [reconstructed in [103]]
[59] Johann Theodor Peters. Zehnstellige Logarithmentafel volume 1 : Zehnstellige Logarithmen von 1 bis 100000 nebst einem Anhang mathematischer Tafeln.
Berlin: Reichsamt f. Landesaufnahme, 1922. [not seen, second edition in 1957 (seen);
also Russian edition in 1964 and perhaps in 1975; reconstructed in [105]; the appendices on mathematical tables are by Peters, J. Stein and G. Witt]
[60] Johann Theodor Peters. Sechsstellige Tafel der trigonometrischen Funktionen : enthaltend die Werte der sechs trigonometrischen Funktionen von zehn zu zehn Bogensekunden des in 90◦ geteilten Quadranten u. d. Werte d. Kotangente u.
Kosekante f. jede Bogensekunde von 0◦ 00 bis 1◦ 200. Berlin: Ferd. Dümmler, 1929. [seen, reprinted in 1939, 1946, 1953, 1962, 1968 and 1971; in Russian in 1975, and perhaps already in 1937 and 1938; reconstructed in [96]]
[61] Johann Theodor Peters. Tafeln zur Verwandlung von rechtwinkligen
Platten-Koordinaten und sphärischen Koordinaten ineinander. Berlin: Ferd.
Dümmler, 1929. [Veröffentlichungen des Astronomischen Rechen-Instituts zu Berlin-Dahlem, number 47]
[62] Johann Theodor Peters. Multiplikations- und Interpolationstafeln für alle ein- bis dreistelligen Zahlen. Berlin: Wichmann, 1930. [reprinted from [63]; reconstructed in [93]]
[63] Johann Theodor Peters. Sechsstellige trigonometrische Tafel für neue Teilung.
Berlin: Wichmann, 1930. [seen, third edition in 1939 and fourth in 1942; an excerpt was reprinted as [62]; reconstructed in [97]]
[64] Johann Theodor Peters. Präzessionstafeln für das Äquinoktium 1950.0. Berlin:
Ferd. Dümmler, 1934. [Veröffentlichungen des Astronomischen Rechen-Instituts zu Berlin-Dahlem, number 50]
[65] Johann Theodor Peters. Tafeln zur Berechnung der jährlichen Präzession in Rektaszension für das Äquinoktium 1950.0. Berlin: Ferd. Dümmler, 1934.
[Veröffentlichungen des Astronomischen Rechen-Instituts zu Berlin-Dahlem, number 51]
17
Evolventen-Funktionen von Hundertstel zu Hundertstel des Grades nebst einigen Hilfstafeln für die Zahnradtechnik. Berlin: Ferd. Dümmler, 1937. [not seen, reprinted in 1951 and 1963 (seen); reconstructed in [101]]
[68] Johann Theodor Peters. Sechsstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades. Berlin: Wichmann, 1938. [seen, 3rd edition in 1940, 5th and 6th in 1942, 7th in 1943, 9th in 1944, 10th in 1953, 12th in 1959, 14th in 1970, and other editions in 1973 and other years; reconstructed in [98]]
[69] Johann Theodor Peters. Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten. Berlin: Verlag des Reichsamts für Landesaufnahme, 1939. [reprinted in 1943 (Ann Arbor, Michigan, perhaps in German, but with an English title) and in 1963, 1965 and 1968 in English under the title “Eight-Place Tables of trigonometric functions for every second of arc.”; the last three editions have [49] as an appendix; there have also been two limited English editions in 1939 and 1940 [5]; the main table was reconstructed in [91]]
[70] Johann Theodor Peters. Siebenstellige Logarithmentafel. Berlin: Verlag des Reichsamts für Landesaufnahme, 1940. [2 volumes, 1: Logarithmen der Zahlen,
Antilogarithmen, etc., 2: Logarithmen der trigonometrischen Funktionen für jede 10. Sekunde d.
Neugrades, etc.; volume 2 is reconstructed in [99]]
[71] Johann Theodor Peters. Siebenstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Neugrades. Berlin: Verlag des Reichsamts für Landesaufnahme, 1941. [reprinted in 1952, 1956 and 1967; reconstructed in [100]]
[72] Johann Theodor Peters, Alfred Lodge, Elsie Jane Ternouth, and Emma Gifford.
Factor table giving the complete decomposition of all numbers less than 100,000.
London: Office of the British Association, 1935. [introduction by Leslie J. Comrie, and bibliography of tables by James Henderson, reprinted in 1963] [reconstructed in [86]]
[73] Johann Theodor Peters and Karl Pilowski. Tafeln zur Berechnung der Präzessionen zwischen den Äquinoktien 1875.0 und 1950.0. Berlin: Ferd.
Dümmler, 1930. [Veröffentlichungen des Astronomischen Rechen-Instituts zu Berlin-Dahlem, number 49]
[74] Johann Theodor Peters and Johannes Stein. Zweiundfünfzigstellige Logarithmen.
Berlin: Ferd. Dümmler, 1919. [Veröffentlichungen des Astronomischen Rechen-Instituts zu Berlin, number 43]
[75] Johann Theodor Peters, Walter Storck, and F. Ludloff. Hütte Hilfstafeln : zur I. Verwandlung von echten Brüchen in Dezimalbrüche ; II. Zerlegung der Zahlen bis 10000 in Primfaktoren ; ein Hilfsbuch zur Ermittelung geeigneter Zähnezahlen für Räderübersetzungen. Berlin: Wilhelm Ernst & Sohn, 1922. [3rd edition]
18
[77] Holger Prytz. Tables d’anti-logarithmes. Copenhague: Lehmann & Stage, 1886.
[not seen]
[78] Denis Roegel. A reconstruction of Adriaan Vlacq’s tables in the Trigonometria artificialis (1633). Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of [118].]
[79] Denis Roegel. A reconstruction of De Decker-Vlacq’s tables in the Arithmetica logarithmica (1628). Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of [117].]
[80] Denis Roegel. A reconstruction of Edward Sang’s table of logarithms (1871).
Technical report, LORIA, Nancy, 2010. [This is a reconstruction of [109].]
[81] Denis Roegel. A reconstruction of Henri Andoyer’s table of logarithms (1911).
Technical report, LORIA, Nancy, 2010. [This is a reconstruction of [2].]
[82] Denis Roegel. A reconstruction of Henri Andoyer’s trigonometric tables
(1915–1918). Technical report, LORIA, Nancy, 2010. [This is a reconstruction of [3].]
[83] Denis Roegel. A reconstruction of the tables of Briggs and Gellibrand’s
Trigonometria Britannica (1633). Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of [20].]
[84] Denis Roegel. A reconstruction of the tables of Briggs’Arithmetica logarithmica (1624). Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of [19].]
[85] Denis Roegel. The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation. Technical report, LORIA, Nancy, 2010.
[86] Denis Roegel. A reconstruction of the table of factors of Peters, Lodge, Ternouth, and Gifford (1935). Technical report, LORIA, Nancy, 2011. [This is a recalculation of the tables of [72].]
[87] Denis Roegel. A reconstruction of Bauschinger and Peters’s eight-place table of logarithms (volume 1, 1910). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [17].]
[88] Denis Roegel. A reconstruction of Bauschinger and Peters’s eight-place table of logarithms (volume 2, 1911). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [17].]
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of [55].]
[91] Denis Roegel. A reconstruction of Peters’s eight-place table of trigonometric functions (1939). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [69].]
[92] Denis Roegel. A reconstruction of Peters’s five-place table of logarithms of trigonometric functions (1912). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [51].]
[93] Denis Roegel. A reconstruction of Peters’s multiplication and interpolation tables (1930). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [62].]
[94] Denis Roegel. A reconstruction of Peters’s seven-place table of logarithms of trigonometric functions (1911). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [50].]
[95] Denis Roegel. A reconstruction of Peters’s seven-place table of trigonometric functions (1918). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [54].]
[96] Denis Roegel. A reconstruction of Peters’s six-place table of trigonometric functions (1929). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [60].]
[97] Denis Roegel. A reconstruction of Peters’s six-place table of trigonometric functions for the new division (1930). Technical report, LORIA, Nancy, 2016.
[This is a reconstruction of [63].]
[98] Denis Roegel. A reconstruction of Peters’s six-place table of trigonometric functions for the new division (1938). Technical report, LORIA, Nancy, 2016.
[This is a reconstruction of [68].]
[99] Denis Roegel. A reconstruction of Peters’s table of 7-place logarithms (volume 2, 1940). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [70].]
[100] Denis Roegel. A reconstruction of Peters’s table of 7-place trigonometrical values for the new division (1941). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [71].]
[101] Denis Roegel. A reconstruction of Peters’s table of involutes (1937). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [67].]
[102] Denis Roegel. A reconstruction of Peters’s table of logarithms to 6 places (1921).
Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [57].]
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[105] Denis Roegel. A reconstruction of Peters’s ten-place table of logarithms (volume 1, 1922). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [59].]
[106] Denis Roegel. A reconstruction of Peters’s ten-place table of logarithms (volume 2, 1919). Technical report, LORIA, Nancy, 2016. [This is a reconstruction of [56].]
[107] Denis Roegel. The genealogy of Johann Theodor Peters’s great mathematical tables. Technical report, LORIA, Nancy, 2016.
[108] Sa. Review of “J. Peters: Achtstellige Tafel der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten”. Astronomische Nachrichten,
269(2):120, 1939. [review of [69]]
[109] Edward Sang. A new table of seven-place logarithms of all numbers from 20 000 to 200 000. London: Charles and Edwin Layton, 1871. [Reconstruction by D. Roegel, 2010 [80].]
[110] Karl Schütte. Index mathematischer Tafelwerke und Tabellen aus allen Gebieten der Naturwissenschaften. München: R. Oldenbourg, 1955.
[111] Dmitri˘ı Selivanov, Julius Bauschinger, and Marie Henri Andoyer. Le calcul des différences et interpolation. In Jules Molk, editor,Encyclopédie des sciences mathématiques pures et appliquées, volume 1(4) (fasc. 1), pages 47–160. Paris:
Gauthier-Villars, 1906. [includes a French edition of [15]]
[112] Daniel Shanks. Jean Peters, Eight-place tables of trigonometric functions for every second of arc. Mathematics of Computation, 18(87):509, 1964. [review of the edition published in 1963 [69]]
[113] Robert Shortrede. Logarithmic tables, to seven places of decimals, containing logarithns to numbers from 1 to 120,000, numbers to logarithms from .0 to 1.00000, logarithmic sines and tangents to every second of the circle, with arguments in space and time, and new astronomical and geodesical tables.
Edinburgh: Adam & Charles Black, 1844. [second edition in 1849]
[114] Friedrich Ludwig Stegmann. Tafel der natürlichen Logarithmen auf fünf
Decimalen und der Gaussischen Logarithmen. Marburg: Joh. Aug. Koch, 1856.
[115] Gustav Stracke. Julius Bauschinger. Monthly Notices of the Royal Astronomical Society, 95(4):336–337, 1935.
[116] John Todd. J. Peters, Ten-place logarithm table. Mathematical Tables and other Aids to Computation, 12:61–63, 1958. [review of the 2nd edition published in 1957 [59, 56]]
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tables were reconstructed by D. Roegel in 2010. [78]]
[119] Stephan Weiss. Die Differenzmaschine von Hamann und die Berechnung der Logarithmen, 2006. www.mechrech.info/publikat/HamDiffM.pdf
[120] Stephan Weiss. Difference engines in the 20th century. In Proceedings 16th International Meeting of Collectors of Historical Calculating Instruments, September 2010, Leiden, pages 157–164, 2010.
[121] Johann Heinrich Westphal. Logarithmische Tafeln. Königsberg:
Universitäts-Buchhandlung, 1821.
[122] Roland Wielen and Ute Wielen. Die Reglements und Statuten des Astronomischen Rechen-Instituts und zugehörige Schriftstücke im Archiv des Instituts. Edition der Dokumente. Heidelberg: Astronomisches Rechen-Institut, 2011. [pp. 255–258 on some archives on Peters]
[123] Roland Wielen and Ute Wielen. Von Berlin über Sermuth nach Heidelberg : Das Schicksal des Astronomischen Rechen-Instituts in der Zeit von 1924 bis 1954 anhand von Schriftstücken aus dem Archiv des Instituts. Heidelberg:
Astronomisches Rechen-Institut, 2012. [various information on Peters, including photographs]
[124] Roland Wielen, Ute Wielen, Herbert Hefele, and Inge Heinrich. Die Geschichte der Bibliothek des Astronomischen Rechen-Instituts. Heidelberg: Astronomisches Rechen-Institut, 2014. [various information on Peters]
[125] Roland Wielen, Ute Wielen, Herbert Hefele, and Inge Heinrich. Supplement zur Geschichte der Bibliothek des Astronomischen Rechen-Instituts. Heidelberg:
Astronomisches Rechen-Institut, 2014. [lists several of Peters’s tables]
[126] Theodor Wittstein. Vierstellige Gaussische Logarithmen in neuer Anordnung.
Astronomische Nachrichten, 51(1208):125–126, 1859.
[127] Theodor Wittstein. Siebenstellige Gaussiche Logarithmen zur Auffindung des Logarithmus der Summe oder Differenz zweier Zahlen, deren Logarithmen gegeben sind. Hannover: Hahn’sche Hofbuchhandlung, 1866. [also with a French title]
[128] Julius Zech. Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen. Leipzig: Weidmann’sche Buchhandlung, 1849.
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